D.I. Survival models and copulas

Transcription

1 D- D. SURVIVAL COPULA D.I. Survival moels an copulas Definiions, relaionships wih mulivariae survival isribuion funcions an relaionships beween copulas an survival copulas. D.II. Fraily moels Use of a laen variable o inrouce epenence beween survival imes. Link wih Archimeean copula D.III. Depenence measures Paricular care shoul be pai when measuring epenence among survival imes. Properies of Kenall s au, Spearman s rho an Tail epenences in a survival seing.

2 D-2 D.IV. Compeing risk moels Definiion an properies D.V. Esimaion Problems of censoring an runcaion. D.VI. Conclusions

3 D-3 D.I. Survival moels an copulas The erm mulivariae survival aa covers he fiel where inepenence beween survival imes canno be assume. We may parallel he consrucion of mulivariae isribuion hrough he use of copulas in a survival framework. Firs we consier he univariae aa separaely in orer o characerize he specific properies of he survival imes. Then we search o escribe he join behavior of he survival imes by aking ino accoun he properies exhibie in he firs sep.

4 D-4 a Univariae survival noions Le T enoe a survival ime wih isribuion F an ensiy f. The survival funcion is given by [ > ] F S P T. The hazar rae or risk funcion λ is efine as λ P[ T + ΔT ] lim Δ 0 Δ. I can be inerpree as he insananeous failure rae assuming he sysem has survive o ime. I is given by f λ. S

5 D-5 The hazar funcion is equal o Λ λ s s. 0 I is also known uner he name: inegrae hazar funcion or cumulaive hazar funcion. We ge he relaionship : S exp Λ In some cases we can incorporae explanaory variables in he moeling of λ, an we have hen λ exp Xβ λ 0 where λ 0 is calle he baseline hazar funcion Cox proporional hazar rae moel.

6 D-6 b Mulivariae survival noions The previous efiniions can be exene o he mulivariae case. The mulivariae survival funcion S is efine by where,..., P T [ > T ] S,..., > T,...,T are survival imes wih S j j. univariae survival funcions We have S0,...,0,,0,...,0 S. j j j Noe ha S,..., F,...,. The ensiy is simply f,...,,..., F,..., S,...,

7 D-7 Mulivariae exensions of he hazar rae an he hazar funcion are given by λ max,..., lim P[ T Δ j 0 Δ... or equivalenly: + Δ Δ,... T,...] λ,..., f S,...,,..., an,...,... s,..., s s Λ λ s Relaionship beween S an Λ canno be simply formulae, since coniional hazar raes nee o be aken ino accoun.

8 D-8 Copulas are hen a naural ools o evelop mulivariae survival funcions from marginal univariae survival funcions. c Survival copulas A mulivariae survival funcion S can be represene as follows : S,..., C S,..., S, where C is a copula Sklar heorem for survival funcions. The survival copula C couples he join survival funcion o is univariae margins in a manner compleely analogous o he way a copula connecs he join isribuion funcion o is margins.

9 There exiss a link beween he survival C an he copula C. In he bivariae case i is given by D-9 C u, u2 u + u2 + C u, u2 Noe ha we can buil a survival funcion as S,..., C S S,..., C S,..., S,..., S or as for a given copula C. This will no yiel he same survival funcions excep in some cases. For example i can be shown ha for ellipical copulas C C normal, suen. I is also rue for he Frank copula. Then i is equivalen o work wih he copula or he survival copula.

10 D-0 D.II. Fraily moels The main iea is o inrouce epenence beween survival imes T,...,T by using an unobserve ranom variable W, calle he fraily. I correspons o a laen or hien variable moeling. Given he fraily W wih isribuion G he survival imes are assume o be inepenen : [ > T > W w] P T,..., j [ > W w] P T j j

11 D- We ake hen S,..., n w j S j W w [ ] j j j w ψ, where ψ j j is he baseline survival funcion in a proporional hazar moel: ψ j j exp Λ exp λ s s i j j 0 i The unconiional join survival funcion is furher efine as S E,..., n [ S,..., ] n W S,..., n w G w We only nee o inegrae w.r.. he isribuion G.

12 D-2 I can be shown ha a survival fraily copula is a special case of he consrucion base on S,..., C S,..., S where C is an Archimeean copula wih a generaor corresponing o he inverse of he Laplace ransform of he isribuion of he fraily variable. Remark ha fraily moels exhibi a PQD behavior only, which migh be an hanicap for he moeling of some aa. Recall ha an Archimeean copula is such ha C u, u2 ϕ ϕ u + ϕ u2 where ϕ is calle he generaor of he copula.

13 D-3 The name Archimeean comes from one of he mahemaical propery of his caegory of copula which is relae o he Archimeean axiom: if a,b are posiive real numbers, hen here exiss an ineger n such ha na>b. Examples are he Frank copula an he Gumbel copula. They fin a wie range of applicaions since hey are easy o consruc, 2 here is a large variey of copula families which belong o his class, 3 hey have nice mahemaical properies. The high egree of analyical racabiliy of he class is an avanage, bu he number of free parameers is ypically low. This migh become an hanicap in high imensions when he epenence srucure of he aa is complex.

14 D-4 D.III. Depenence measures a linear correlaion The raiional way of evaluaing epenence in a bivariae isribuion is by means of he sanar correlaion coefficien. This measure of epenence is naural an unproblemaic in he class of ellipical isribuions, bu i migh be misleaing in oher conexs, ypically encounere in survival aa. Here are some usual misinerpreaions of he Pearson correlaion couner-examples may be given.. T an T 2 are inepenen if an only if T, T 0. corr 2 2. corr T, T2 0 means ha here is no perfec epenence beween T an T 2.

15 D-5 3. for given margins, he permissible range of T, is [-,]. corr T 2 Survival aa are ypically posiive. Hence he lower boun can never been reache. I is furher ifficul o obain large range of correlaion because of he ype of isribuions generally use in survival moeling. For he Weibull, he inerval is ofen [-/3,/2] only. b Kenall s au an Spearman s rho The Kenall s au an Spearman s rho of he survival copula an is associae copula are equal.

16 D-6 c Tail epenence Tail epenence measures correspon o Upper ail epenence: [ > uu u] λ U lim P U 2 > u If λ U 0, ], hen upper ail epenence. If λ 0, hen no upper ail epenence. U Lower ail epenence: [ < uu u] λ L lim P U 2 < u 0 If λ L 0,], hen lower ail epenence. If λ 0, hen no lower ail epenence. L The upper ail epenence of he survival copula will give he lower ail epenence of is associae copula, an vice-versa.

17 D-7 Lower ail epenence in survival copula will characerize immeiae join eah, while upper ail epenence in survival copula will characerize long-erm join survival. Remark: Normal copula has no upper or lower ail epenence. Suen copula may. D.IV. Compeing risk moels Compeing risk moels correspon o he suy of any failure process in which here are ifferen causes of failures. T,...,T Le us consier survival imes. In a compeing risk moel he survival ime τ is efine by τ min T,..., T.

18 D-8 We have hen [ ],...,,..., min S S C T T P S τ The cf of he survival ime τ is,...,,..., F F C S S C F τ an is ensiy is given by,..., f S S C f i i i τ Explici forms can be foun for example for Weibull margins an a Gumbel copulas.

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21 D-2 Uner an ii scheme we ge F F τ an f τ F f D.V. Esimaion The esimaion by maximum likelihoo are exacly he same as before when observaions are complee. Inee ML esimaion relies on he join ensiy of he survival imes. However ealing wih survival imes is no as simple, because recors on survival rais are ofen incomplee: survival aa are ofen censore or runcae.

22 D-22 Uner lef runcaion we only observe aa above a fixe hreshol. We have no informaion abou he behavior below he limi only repore losses above a given level. Uner censoring we have usually a mixure beween complee an incomplee aa. For example uner righ censoring we observe T if i is below a hreshol C or he hreshol C iself if i is above. The hreshol C may be fixe or ranom. Esimaion uner hese schemes are much more ifficul, especially when ealing wih nonparameric esimaion. For example uner lef runcaion i is impossible o ienify nonparamerically he par of he isribuion below he hreshol we have no informaion!.

23 D-23 D.VI. Conclusions The join behavior of survival imes can be easily moele hrough copulas. I is a powerful ool o analyze he epenence srucure among hese aa, especially because symmeric isribuions are no naural caniaes for hese aa. Esimaion proceures are also available in such a seing bu are more ifficul o implemen when censoring or runcaion mechanisms are presen.